Dispersion Relation

The package is located at mma/dispersion-relation.wl. The corresponding test notebook is located at mma/package-test/dispersion-relation-package-test.nb.

For a reference of paper about this topic, please read

  1. Izaguirre, I., Raffelt, G., & Tamborra, I. (2017). Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion Relation Approach. Physical Review Letters, 118(2), 21101.

Two Beams

N Beams

Box Spectra

In the package, a box spectrum is defined as

\[\{ \{ \{ u_1, u_1' \}, g_1 \} ,\{ \{ u_2, u_2' \}, g_2 \}, ,\{ \{ u_3, u_3' \}, g_3 \} , \cdots \},\]

where \(u_i\) is the start \(\cos\theta_i\) value and \(u_i'\) is the ending value of \(\cos\theta_i\), within these two values, we have the spectrum value \(g_1\).

The functions defined in this section can take in spectrum of arbitrary segments.

During the calculation of any quantities in this problem, the integral

\[I_m = \int_{c_1}^{c_2} \frac{u^m}{1-n u} \, du\]

is widely used. These integrals can be calculated analytically.

IntFun0n[n,c1,c2]

Calculates the value of \(I_0\) for given \(n\), \(c_1\), and \(c_2\).

Parameters:
  • n – the variable \(n\)
  • c1 – the lower limit of the integral
  • c2 – the upper limit of the integral
Return type:

a real or complex number
IntFun1n[n,c1,c2]

Calculates the value of \(I_1\) for given \(n\), \(c_1\), and \(c_2\).

Parameters:
  • n – the variable \(n\)
  • c1 – the lower limit of the integral
  • c2 – the upper limit of the integral
Return type:

a real or complex number
IntFun2n[n,c1,c2]

Calculates the value of \(I_2\) for given \(n\), \(c_1\), and \(c_2\).

Parameters:
  • n – the variable \(n\)
  • c1 – the lower limit of the integral
  • c2 – the upper limit of the integral
Return type:

a real or complex number
ConAxialSymOmegaNMAA[n,spect_optional]

Calculates \(\omega(n)\) for MAA solution for given spectrum.

Parameters:
  • n – the variable \(n\)
  • spect – the input spectrum, which is optional. The default spectrum is {{{0.3,0.6},1},{{0.6,0.9},-1}}
Return type:

real or complex number
ConAxialSymOmegaNMZA[n,spect_optional]

Calculates \(\omega(n)\) for MZA solutions for given spectrum.

Parameters:
  • n – the variable \(n\)
  • spect – the input spectrum, which is optional. The default spectrum is {{{0.3,0.6},1},{{0.6,0.9},-1}}
Return type:

a list of real or complex number, {MZA+, MZA-}